Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

Some new relations on skew Schur function differences are established both combinatorially using Schützenberger’s jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain differences of skew Schur functions are Schur positive. Applying these results to a basis of symmetric functions involving ribbon Schur functions confirms the validity of a Schur positivity conjecture due to McNamara. A further application reveals that certain differences of products of Schubert classes are Schubert positive. For Manfred Schocker 1970–2006. S.J. van Willigenburg was supported in part by the National Sciences and Engineering Research Council of Canada.     

A Longitudinal Study of Elementary Pre‐service Teachers' Mathematics Beliefs and Content Knowledge

This study investigated the mathematics beliefs and content knowledge of 103 elementary pre‐service teachers in a developmental teacher preparation program that included a two course mathematics methods sequence. Pre‐service teachers' pedagogical beliefs became more cognitively‐oriented during the teacher preparation program with these changes occurring during the two methods courses. Pedagogical beliefs remained stable during student teaching. The pre‐service teachers also significantly increased their personal efficacy for teaching mathematics throughout the program with these shifts occurring across both methods courses and into student teaching. Pedagogical beliefs and teaching efficacy beliefs were not related at the beginning of the program, but, in general, were positively related throughout the program. In addition, the pre‐service teachers' pedagogical beliefs were positively related to their specialized content knowledge for teaching mathematics at the end of the program.     

Imine coordinated 2-aminoazole complexes of (CO)5Cr(0) and (CO)5W(0), verification by structural characterisation

The synthesis and structural characterisation (IR, MS, ¹H, ¹³C and ¹⁵N NMR and single crystal X-ray diffraction analysis) of 2-aminoazole complexes of pentacarbonyl Cr(0) and W(0) are described. When provided with endocyclic soft thioether or endocyclic hard amine, endocyclic borderline imine and exocyclic hard amine coordination sites, the softer endocyclic imine coordination site is favoured.     

Cyclically permutable representations of cyclic codes

Cyclically permutable codes have been studied for several applications involving synchronization, code-division multiple-access (CDMA) radio systems and optical CDMA. The usual emphasis is on finding constant weight cyclically permutable codes with the maximum number of codewords. In this paper the question of when a particular error-correcting code is equivalent (by permutation of the symbols) to a cyclically permutable code is addressed. The problem is introduced for simplex codes and a motivating example is given. In the final section it is shown that the construction technique may be applied in general to cyclic codes.     

Distinction and quantification of inorganic sulfur species including thiosulfate by X‐ray fluorescence (WD‐XRF)

Sulfur occurs in a variety of inorganic and organic compounds with oxidation states from ?II up to +VI. Differentiation of these species in solid geochemical samples can be challenging because of oxidation processes during sample preparation by acidic digestion. Applying pressed powder pellets and an analysis by wavelength‐dispersive X‐ray fluorescence minimises reactions with oxidants and water. Main subjects of this work were five inorganic sulfur species, sulfide ?II, elemental sulfur 0, thiosulfate +II, sulfite +IV, and sulfate +VI, and the determination of their fluorescence energies in the sulfur X‐ray spectra. S Kα_1,2 and S Kβ₁ can be observed for all species, S Kβ′ satellites only for species with coordinated oxygen. The results are in good agreement with previously published data. Yet none of the 38 investigated papers from the past 90 years reported S Kα_1,2 of thiosulfate, which was determined as E = 2,309.12 eV in this work apparently for the first time. Binary mixtures of sulfur species are strongly differing in their ability of being quantitatively differentiated, as a reliable quantification requires a sufficient difference of the respective fluorescence energies. Regression equations for each mixture can be used to calculate the ratio of mass fractions of the investigated species from the evaluated fluorescence energy. If boundary conditions are considered, the presented approaches can be applied for analyses of geochemical samples or quality control of technical products. The main advantage of the described methods is the option of implementation to everyday X‐ray fluorescence lab routine without substantial additional effort.     

Impact of Xylose on Dynamics of Water Diffusion in Mesoporous Zeolites Measured by NMR

Zeolites are known to be effective catalysts in biomass converting processes. Understanding the mesoporous structure and dynamics within it during such reactions is important in effectively utilizing them. Nuclear magnetic resonance (NMR) T₂ relaxation and diffusion measurements, using a high-power radio frequency probe, are shown to characterize the dynamics of water in mesoporous commercially made 5A zeolite beads before and after the introduction of xylose. Xylose is the starting point in the dehydration into furfural. The results indicate xylose slightly enhances rotational mobility while it decreases translational motion through altering the permeability, K, throughout the porous structure. The measurements show xylose inhibits pure water from relocating into larger pores within the zeolite beads where it eventually is expelled from the bead itself.